In this section we provide a simpler sufficient condition for (REF ) to hold
for GWF models whose genealogies converge to a \(\Lambda \) -coalescent. There are two simple heuristics from distinct population dynamics
which motivate this section: 1) all the individuals in the population have the same
fitness but occasionally, by mere chance, there occur large reproductive events where a single
individual takes up a large fraction of the offspring in the next generation, and
2) the offspring distribution is highly asymmetric
and the family frequencies given by the mass-partition \(\tilde{\rho }^{(N)}\) 
are dominated by the family frequency \(\tilde{\rho }^{(N)}_1\)  of the individuals with the highest reproductive success.
In both of these scenarios
we expect that all the coagulations that remain observable in the limit correspond to very large reproduction
events of single individuals along the generations (see [1]}, [2]}, [3]});
these coagulations must then be driven by \(\tilde{\rho }^{(N)}_1\) , the largest value of
\(\tilde{\rho }^{(N)}\) . In practice we would rather allow more flexibility and consider any random (possibly unordered) vector
of probabilities
\(\eta ^{(N)}{\in }\Delta _N\) 
and prove that all the coagulations that “survive” in the limit are driven by the frequency
\(\eta ^{(N)}_{1}\) . This entails that \(\eta ^{(N)}_{1}\)  should approximate \(\tilde{\rho }^{(N)}_1\)  as \(N\rightarrow \infty \) , but working
with \(\eta ^{(N)}_{1}\)  instead of \(\tilde{\rho }^{(N)}_1\)  may ease the study of the underlying genealogy of the population
(also note that our setup does include the case
where \(\eta ^{(N)}_i=\tilde{\rho }^{(N)}_i\)  for all \(1\le i\le N\) ). Of particular interest is the case where \(\left(\eta ^{(N)}_{1},\dots ,\eta ^{(N)}_{N}\right)\)  is a
(normalized) size-biased pick from \(\tilde{\rho }^{(N)}\)  for which the distribution may be explicitly known (see Section  for a more general example of this).
